S3 BIC method for class "GEM"

Description

S3 BIC method for objects of class GEM.

Usage

## S3 method for class 'GEM'
BIC(object, ...)

Arguments

Value

BIC index for the fitted model.

See Also

logLik, GEM

Main function for CUB models

Description

Main function to estimate and validate a CUB model for explaining uncertainty and feeling for given ratings, with or without covariates and shelter effect.

Usage

CUB(Formula, data, ...)

Arguments

Details

This is the main function for CUB models, which calls for the corresponding functions whenever covariates or shelter effect are specified. It performs maximum likelihood estimation via the E-M algorithm for CUB models and extensions. The optimization procedure is run via "optim".
It is possible to fit data with CUB models, with or without covariates for the parameters of the mixture model, and CUB models with shelter effect with no covariate included in the model. The program also checks if the estimated variance-covariance matrix is positive definite: if not, it prints a warning message and returns a matrix and related results with NA entries.

Value

An object of the class "GEM"-CUB": returns a list containing the following results:

References

Piccolo D. and D'Elia A. (2008). A new approach for modelling consumers' preferences, Food Quality and Preference, 18, 247–259
Iannario M. and Piccolo D. (2012). CUB models: Statistical methods and empirical evidence, in: Kenett R. S. and Salini S. (eds.), Modern Analysis of Customer Surveys: with applications using R, J. Wiley and Sons, Chichester, 231–258
Iannario M. (2012). Modelling shelter choices in a class of mixture models for ordinal responses, Statistical Methods and Applications, 21, 1–22
Iannario M. and Piccolo D. (2014). Inference for CUB models: a program in R, Statistica & Applicazioni, XII n.2, 177–204
Iannario M. (2016). Testing the overdispersion parameter in CUBE models, Communications in Statistics: Simulation and Computation, 45(5), 1621–1635

See Also

probcub00, probcubp0, probcub0q, probcubpq, probcubshe1, loglikCUB, varmatCUB

Main function for CUBE models

Description

Main function to estimate and validate a CUBE model for given ratings, explaining uncertainty, feeling and overdispersion.

Usage

CUBE(Formula,data,...)

Arguments

Details

It is the main function for CUBE models, calling for the corresponding functions whenever covariates are specified: it is possible to select covariates for explaining all the three parameters or only the feeling component.
The program also checks if the estimated variance-covariance matrix is positive definite: if not, it prints a warning message and returns a matrix and related results with NA entries. The optimization procedure is run via "optim". If covariates are included only for feeling, the variance-covariance matrix is computed as the inverse of the returned numerically differentiated Hessian matrix (option: hessian=TRUE as argument for "optim"), and the estimation procedure is not iterative, so a NULL result for $niter is produced. If the estimated variance-covariance matrix is not positive definite, the function returns a warning message and produces a matrix with NA entries.

Value

An object of the class "GEM"-"CUBE" is a list containing the following results:

References

Iannario M. (2014). Modelling Uncertainty and Overdispersion in Ordinal Data, Communications in Statistics - Theory and Methods, 43, 771–786
Piccolo D. (2015). Inferential issues for CUBE models with covariates, Communications in Statistics. Theory and Methods, 44(23), 771–786.
Iannario M. (2015). Detecting latent components in ordinal data with overdispersion by means of a mixture distribution, Quality & Quantity, 49, 977–987
Iannario M. (2016). Testing the overdispersion parameter in CUBE models. Communications in Statistics: Simulation and Computation, 45(5), 1621–1635.

See Also

probcube, loglikCUBE, loglikcuben, inibestcube, inibestcubecsi, inibestcubecov, varmatCUBE

CUB package

Description

The analysis of human perceptions is often carried out by resorting to questionnaires, where respondents are asked to express ratings about the items being evaluated. The standard goal of the statistical framework proposed for this kind of data (e.g. cumulative models) is to explicitly characterize the respondents' perceptions about a latent trait, by taking into account, at the same time, the ordinal categorical scale of measurement of the involved statistical variables.
The new class of models starts from a particular assumption about the unconscious mechanism leading individuals' responses to choose an ordinal category on a rating scale. The basic idea derives from the awareness that two latent components move the psychological process of selection among discrete alternatives: attractiveness towards the item and uncertainty in the response. Both components of models concern the stochastic mechanism in term of feeling, which is an internal/personal movement of the subject towards the item, and uncertainty pertaining to the final choice.
Thus, on the basis of experimental data and statistical motivations, the response distribution is modelled as the convex Combination of a discrete Uniform and a shifted Binomial random variable (denoted as CUB model) whose parameters may be consistently estimated and validated by maximum likelihood inference. In addition, subjects' and objects' covariates can be included in the model in order to assess how the characteristics of the respondents may affect the ordinal score.
CUB models have been firstly introduced by Piccolo (2003) and implemented on real datasets concerning ratings and rankings by D'Elia and Piccolo (2005).
The CUB package allows the user to estimate and test CUB models and their extensions by using maximum likelihood methods: see Piccolo and Simone (2019a, 2019b) for an updated overview of methodological developments and applications. The accompanying vignettes supplies the user with detailed usage instructions and examples.
Acknowledgements: The Authors are grateful to Maria Antonietta Del Ferraro, Francesco Miranda and Giuseppe Porpora for their preliminary support in the implementation of the first version of the package.

Details

Author(s)

Maria Iannario, Domenico Piccolo, Rosaria Simone

References

D'Elia A. (2003). Modelling ranks using the inverse hypergeometric distribution, Statistical Modelling: an International Journal, 3, 65–78
Piccolo D. (2003). On the moments of a mixture of uniform and shifted binomial random variables, Quaderni di Statistica, 5, 85–104
D'Elia A. and Piccolo D. (2005). A mixture model for preferences data analysis, Computational Statistics & Data Analysis, 49, 917–937
Piccolo D. and Simone R. (2019a). The class of CUB models: statistical foundations, inferential issues and empirical evidence. Statistical Methods and Applications, 28(3), 389–435.
Piccolo D. and Simone R. (2019b). Rejoinder to the discussions: The class of CUB models: statistical foundations, inferential issues and empirical evidence. Statistical Methods and Applications, 28(3), 477-493.
Capecchi S. and Piccolo D. (2017). Dealing with heterogeneity in ordinal responses, Quality and Quantity, 51(5), 2375–2393
Metron, 74(2), 233–252.
Iannario M. and Piccolo D. (2016b). A generalized framework for modelling ordinal data. Statistical Methods and Applications, 25, 163–189.

Main function for CUSH models

Description

Main function to estimate and validate a CUSH model for ordinal responses, with or without covariates to explain the shelter effect.

Usage

CUSH(Formula,data,...)

Arguments

Details

The estimation procedure is not iterative, so a null result for CUSH$niter is produced. The optimization procedure is run via "optim". If covariates are included, the variance-covariance matrix is computed as the inverse of the returned numerically differentiated Hessian matrix (option: hessian=TRUE as argument for "optim"). If not positive definite, it returns a warning message and produces a matrix with NA entries.

Value

An object of the class "CUSH" is a list containing the following results:

References

Capecchi S. and Piccolo D. (2015). Dealing with heterogeneity/uncertainty in sample survey with ordinal data, IFCS Proceedings, University of Bologna
Capecchi S. and Iannario M. (2016). Gini heterogeneity index for detecting uncertainty in ordinal data surveys, Metron - DOI: 10.1007/s40300-016-0088-5

See Also

loglikCUSH

Main function for GEM models

Description

Main function to estimate and validate GEneralized Mixture models with uncertainty.

Usage

GEM(Formula,family=c("cub","cube","ihg","cush"),data,...)

Arguments

Details

It is the main function for GEM models estimation, calling for the corresponding function for the specified subclass. The number of categories m is internally retrieved but it is advisable to pass it as an argument to the call if some category has zero frequency.
If family="cub", then a CUB mixture model is fitted to the data to explain uncertainty, feeling and possible shelter effect by further passing the extra argument shelter for the corresponding category. Subjects' covariates can be included by specifying covariates matrices in the Formula as ordinal~Y|W|X, to explain uncertainty (Y), feeling (W) or shelter (X). Notice that covariates for shelter effect can be included only if specified for both feeling and uncertaint (GeCUB models).
If family="cube", then a CUBE mixture model (Combination of Uniform and Beta-Binomial) is fitted to the data to explain uncertainty, feeling and overdispersion. Subjects' covariates can be also included to explain the feeling component or all the three components by specifying covariates matrices in the Formula as ordinal~Y|W|Z to explain uncertainty (Y), feeling (W) or overdispersion (Z). An extra logical argument expinform indicates whether or not to use the expected or the observed information matrix (default is FALSE).
If family="ihg", then an IHG model is fitted to the data. IHG models (Inverse Hypergeometric) are nested into CUBE models (see the references below). The parameter \theta gives the probability of observing the first category and is therefore a direct measure of preference, attraction, pleasantness toward the investigated item. This is the reason why \theta is customarily referred to as the preference parameter of the IHG model. Covariates for the preference parameter \theta have to be specified in matrix form in the Formula as ordinal~U.
If family="cush", then a CUSH model is fitted to the data (Combination of Uniform and SHelter effect). The category corresponding to the inflation should be passed via argument shelter. Covariates for the shelter parameter \delta are specified in matrix form Formula as ordinal~X.
Even if no covariate is included in the model for a given component, the corresponding model matrix needs always to be specified: in this case, it should be set to 0 (see examples below). Extra arguments include the maximum number of iterations (maxiter, default: maxiter=500) for the optimization algorithm and the required error tolerance (toler, default: toler=1e-6).
Standard methods: logLik(), BIC(), vcov(), fitted(), coef(), print(), summary() are implemented.
The optimization procedure is run via optim() when required. If the estimated variance-covariance matrix is not positive definite, the function returns a warning message and produces a matrix with NA entries.

Value

An object of the class "GEM" is a list containing the following elements:

References

D'Elia A. (2003). Modelling ranks using the inverse hypergeometric distribution, Statistical Modelling: an International Journal, 3, 65–78
D'Elia A. and Piccolo D. (2005). A mixture model for preferences data analysis, Computational Statistics & Data Analysis, 49, 917–937
Capecchi S. and Piccolo D. (2017). Dealing with heterogeneity in ordinal responses, Quality and Quantity, 51(5), 2375–2393
Iannario M. (2014). Modelling Uncertainty and Overdispersion in Ordinal Data, Communications in Statistics - Theory and Methods, 43, 771–786
Piccolo D. (2015). Inferential issues for CUBE models with covariates, Communications in Statistics. Theory and Methods, 44(23), 771–786.
Iannario M. (2015). Detecting latent components in ordinal data with overdispersion by means of a mixture distribution, Quality & Quantity, 49, 977–987
Iannario M. and Piccolo D. (2016a). A comprehensive framework for regression models of ordinal data. Metron, 74(2), 233–252.
Iannario M. and Piccolo D. (2016b). A generalized framework for modelling ordinal data. Statistical Methods and Applications, 25, 163–189.

See Also

logLik, coef, BIC, makeplot, summary, vcov, fitted, cormat

Examples

library(CUB)
## CUB models with no covariates
model<-GEM(Formula(Walking~0|0|0),family="cub",data=relgoods)
coef(model,digits=5)     # Estimated parameter vector (pai,csi)
logLik(model)            # Log-likelihood function at ML estimates
vcov(model,digits=4)     # Estimated Variance-Covariance matrix
cormat(model)            # Parameter Correlation matrix
fitted(model)            # Fitted probability distribution
makeplot(model)
################
## CUB model with shelter effect
model<-GEM(Formula(officeho~0|0|0),family="cub",shelter=7,data=univer)
BICshe<-BIC(model,digits=4)
################
## CUB model with covariate for uncertainty
modelcovpai<-GEM(Formula(Parents~Smoking|0|0),family="cub",data=relgoods)
fitted(modelcovpai)
makeplot(modelcovpai)
################
## CUB model with covariates for both uncertainty and feeling components
data(univer)
model<-GEM(Formula(global~gender|freqserv|0),family="cub",data=univer,maxiter=50,toler=1e-2)
param<-coef(model)
bet<-param[1:2]      # ML estimates of coefficients for uncertainty covariate: gender
gama<-param[3:4]     # ML estimates of coefficients for feeling covariate: lage
##################
## CUBE models with no covariates
model<-GEM(Formula(MeetRelatives~0|0|0),family="cube",starting=c(0.5,0.5,0.1),
  data=relgoods,expinform=TRUE,maxiter=50,toler=1e-2)
coef(model,digits=4)       # Final ML estimates
vcov(model)
fitted(model)
makeplot(model)
summary(model)
##################
## IHG with covariates
modelcov<-GEM(willingn~freqserv,family="ihg",data=univer)
omega<-coef(modelcov)      ## ML estimates 
maxlik<-logLik(modelcov)   ## 
makeplot(modelcov)
summary(modelcov)
###################
## CUSH models without covariate
model<-GEM(Dog~0,family="cush",shelter=1,data=relgoods)
delta<-coef(model)      # ML estimates of delta
maxlik<-logLik(model)   # Log-likelihood at ML estimates
summary(model)
makeplot(model)

Hadamard product of a matrix with a vector

Description

Return the Hadamard product between the given matrix and vector: this operation corresponds to multiply every row of the matrix by the corresponding element of the vector, and it is equivalent to the standard matrix multiplication to the right with the diagonal matrix whose diagonal is the given vector. It is possible only if the length of the vector equals the number of rows of the matrix, otherwise it prints an error message.

Usage

Hadprod(Amat, xvett)

Arguments

Details

It is an auxiliary function needed for computing the variance-covariance matrix of the estimated model with covariates.

Value

A matrix of the same dimensions as Amat

Main function for IHG models

Description

Main function to estimate and validate an Inverse Hypergeometric model, without or with covariates for explaining the preference parameter.

Usage

IHG(Formula, data, ...)

Arguments

Details

This is the main function for IHG models (that are nested into CUBE models, see the references below), calling for the corresponding function whenever covariates are specified.
The parameter \theta represents the probability of observing a rating corresponding to the first category and is therefore a direct measure of preference, attraction, pleasantness toward the investigated item. This is reason why \theta is customarily referred to as the preference parameter of the IHG model.
The estimation procedure is not iterative, so a null result for IHG$niter is produced.
The optimization procedure is run via "optim". The variance-covariance matrix (or the estimated standard error for theta if no covariate is included) is computed as the inverse of the returned numerically differentiated Hessian matrix (option: hessian=TRUE as argument for optim). If not positive definite, it returns a warning message and produces a matrix with NA entries.

Value

An object of the class "IHG" is a list containing the following results:

References

D'Elia A. (2003). Modelling ranks using the inverse hypergeometric distribution, Statistical Modelling: an International Journal, 3, 65–78
Iannario M. (2012). CUBE models for interpreting ordered categorical data with overdispersion, Quaderni di Statistica, 14, 137–140

See Also

probihg, iniihg, loglikIHG

Auxiliary function for the log-likelihood estimation of GeCUB models.

Description

Define the opposite one of the two scalar functions that are maximized when running the E-M algorithm for GeCUB models with covariates for feeling, uncertainty and overdispersion.

Usage

Q2gecub(param,datidue)

Arguments

Auxiliary function for the log-likelihood estimation of CUBE models with covariates

Description

Define the opposite of one of the two scalar functions that are maximized when running the E-M algorithm for CUBE models with covariates for feeling, uncertainty and overdispersion.

Usage

Qdue(param, esterno2, q, m)

Arguments

Details

It is iteratively called as an argument of "optim" within CUBE function (with covariates) as the function to minimize to compute the maximum likelihood estimates for the feeling and the overdispersion components.

Auxiliary function for the log-likelihood estimation of CUBE models with covariates

Description

Define the opposite one of the two scalar functions that are maximized when running the E-M algorithm for CUBE models with covariates for feeling, uncertainty and overdispersion.

Usage

Quno(bet, esterno1)

Arguments

Details

It is iteratively called as an argument of "optim" within CUBE function (with covariates) as the function to minimize to compute the maximum likelihood estimates for the feeling and the overdispersion components.

Auxiliary function for the log-likelihood estimation of GeCUB models.

Description

Define the opposite one of the two scalar functions that are maximized when running the E-M algorithm for GeCUB models with covariates for feeling, uncertainty and overdispersion.

Usage

Qunogecub(param,datiuno,s)

Arguments

Auxiliary matrix

Description

Returns an auxiliary matrix needed for computing the variance-covariance matrix of a CUBE model with covariates.

Usage

auxmat(m, vettcsi, vettphi, a, b, c, d, e)

Arguments

References

Iannario, M. (2014). Modelling Uncertainty and Overdispersion in Ordinal Data, Communications in Statistics- Theory and Methods, 43, 771–786
Piccolo, D. (2014). Inferential issues on CUBE models with covariates, Communications in Statistics. Theory and Methods, 44, DOI: 10.1080/03610926.2013.821487

Beta-Binomial probabilities of ordinal responses, with feeling and overdispersion parameters for each observation

Description

Compute the Beta-Binomial probabilities of ordinal responses, given feeling and overdispersion parameters for each observation.

Usage

betabinomial(m,ordinal,csivett,phivett)

Arguments

Details

The Beta-Binomial distribution is the Binomial distribution in which the probability of success at each trial is random and follows the Beta distribution. It is frequently used in Bayesian statistics, empirical Bayes methods and classical statistics as an overdispersed binomial distribution.

Value

A vector of the same length as ordinal, containing the Beta-Binomial probabilities of each observation, for the corresponding feeling and overdispersion parameters.

References

Iannario, M. (2014). Modelling Uncertainty and Overdispersion in Ordinal Data, Communications in Statistics - Theory and Methods, 43, 771–786
Piccolo D. (2015). Inferential issues for CUBE models with covariates. Communications in Statistics - Theory and Methods, 44(23), 771–786.

See Also

betar, betabinomialcsi

Examples

data(relgoods)
m<-10
ordinal<-relgoods$Tv
age<-2014-relgoods$BirthYear
no_na<-na.omit(cbind(ordinal,age))
ordinal<-no_na[,1]; age<-no_na[,2]
lage<-log(age)-mean(log(age))
gama<-c(-0.6, -0.3)
csivett<-logis(lage,gama)
alpha<-c(-2.3,0.92); 
ZZ<-cbind(1,lage)
phivett<-exp(ZZ%*%alpha)
pr<-betabinomial(m,ordinal,csivett,phivett)
plot(density(pr))

Beta-Binomial probabilities of ordinal responses, given feeling parameter for each observation

Description

Compute the Beta-Binomial probabilities of given ordinal responses, with feeling parameter specified for each observation, and with the same overdispersion parameter for all the responses.

Usage

betabinomialcsi(m,ordinal,csivett,phi)

Arguments

Value

A vector of the same length as ordinal: each entry is the Beta-Binomial probability for the given observation for the corresponding feeling and overdispersion parameters.

References

Iannario, M. (2014). Modelling Uncertainty and Overdispersion in Ordinal Data, Communications in Statistics - Theory and Methods, 43, 771–786
Piccolo D. (2015). Inferential issues for CUBE models with covariates. Communications in Statistics - Theory and Methods, 44(23), 771–786.

See Also

betar, betabinomial

Examples

data(relgoods)
m<-10
ordinal<-relgoods$Tv
age<-2014-relgoods$BirthYear
no_na<-na.omit(cbind(ordinal,age))
ordinal<-no_na[,1]; age<-no_na[,2]
lage<-log(age)-mean(log(age))
gama<-c(-0.61,-0.31)
phi<-0.16 
csivett<-logis(lage,gama)
pr<-betabinomialcsi(m,ordinal,csivett,phi)
plot(density(pr))

Beta-Binomial distribution

Description

Return the Beta-Binomial distribution with parameters m, csi and phi.

Usage

betar(m,csi,phi)

Arguments

Value

The vector of length m of the Beta-Binomial distribution.

References

Iannario, M. (2014). Modelling Uncertainty and Overdispersion in Ordinal Data, Communications in Statistics - Theory and Methods, 43, 771–786

See Also

betabinomial

Examples

m<-9
csi<-0.8
phi<-0.2
pr<-betar(m,csi,phi)
plot(1:m,pr,type="h", main="Beta-Binomial distribution",xlab="Ordinal categories")
points(1:m,pr,pch=19)

Shifted Binomial probabilities of ordinal responses

Description

Compute the shifted Binomial probabilities of ordinal responses.

Usage

bitcsi(m,ordinal,csi)

Arguments

Value

A vector of the same length as ordinal, where each entry is the shifted Binomial probability of the corresponding observation.

References

Piccolo D. (2003). On the moments of a mixture of uniform and shifted binomial random variables, Quaderni di Statistica, 5, 85–104

See Also

probcub00, probcubp0, probcub0q

Examples

data(univer)
m<-7
csi<-0.7
ordinal<-univer$informat
pr<-bitcsi(m,ordinal,csi)

Shifted Binomial distribution with covariates

Description

Return the shifted Binomial probabilities of ordinal responses where the feeling component is explained by covariates via a logistic link.

Usage

bitgama(m,ordinal,W,gama)

Arguments

Value

A vector of the same length as ordinal, where each entry is the shifted Binomial probability for the corresponding observation and feeling value.

See Also

logis, probcub0q, probcubpq

Examples

n<-100
m<-7
W<-sample(c(0,1),n,replace=TRUE)
gama<-c(0.2,-0.2)
csivett<-logis(W,gama)
ordinal<-rbinom(n,m-1,csivett)+1
pr<-bitgama(m,ordinal,W,gama)

Pearson X^2 statistic

Description

Compute the X^2 statistic of Pearson for CUB models with one or two discrete covariates for the feeling component.

Usage

chi2cub(m,ordinal,W,pai,gama)

Arguments

Details

No missing value should be present neither for ordinal nor for covariate matrices: thus, deletion or imputation procedures should be preliminarily run.

Value

A list with the following components:

References

Tutz, G. (2012). Regression for Categorical Data, Cambridge University Press, Cambridge

Examples

data(univer)
m<-7
pai<-0.3
gama<-c(0.1,0.7)
ordinal<-univer$informat; W<-univer$gender;
pearson<-chi2cub(m,ordinal,W,pai,gama)
degfree<-pearson$df
statvalue<-pearson$chi2
deviance<-pearson$dev

Pearson X^2 statistic for CUB models with one discrete covariate for feeling

Description

Compute the X^2 statistic of Pearson for the goodness of fit of a CUB model for ordinal responses, where the feeling parameter is explained via a logistic transform of the only discrete covariate. It groups ratings in classes according to the values of the covariate.

Usage

chi2cub1cov(m, ordinal, covar, pai, gama)

Arguments

Value

It returns the following results in a list:

References

Tutz, G. (2011). Regression for categorical data, Cambridge Series in Statistical and Probabilistic Mathematics

Pearson X^2 statistic for CUB models with one discrete covariate for feeling

Description

Compute the X^2 statistic of Pearson for the goodness of fit of a CUB model for ordinal responses, where the feeling parameter is explained via a logistic transform of the only discrete covariate. It groups ratings in classes according to the values of the covariate.

Usage

chi2cub2cov(m, ordinal, covar1, covar2, pai, gama)

Arguments

Value

It returns the following results in a list:

References

Tutz, G. (2011). Regression for categorical data, Cambridge Series in Statistical and Probabilistic Mathematics

S3 Method: coef for class "GEM"

Description

S3 method: coef for objects of class GEM.

Usage

## S3 method for class 'GEM'
coef(object, ...)

Arguments

Details

Returns estimated values of coefficients of the fitted model

Value

ML estimates of parameters of the fitted GEM model.

See Also

GEM, summary

Correlation matrix for estimated model

Description

Compute parameter correlation matrix for estimated model as returned by an object of class "GEM".

Usage

cormat(object,digits=options()$digits)

Arguments

Value

Parameters correlation matrix for fitted GEM models.

See Also

GEM, vcov

Main function for CUB models without covariates

Description

Function to estimate and validate a CUB model without covariates for given ordinal responses.

Usage

cub00(m, ordinal, maxiter, toler)

Arguments

Value

An object of the class "CUB"

See Also

CUB, probbit, probcub00, loglikCUB

Main function for CUB models with covariates for the feeling component

Description

Function to estimate and validate a CUB model for given ordinal responses, with covariates for explaining the feeling component.

Usage

cub0q(m, ordinal, W, maxiter, toler)

Arguments

Value

An object of the class "CUB"

References

Piccolo D. and D'Elia A. (2008), A new approach for modelling consumers' preferences, Food Quality and Preference, 18, 247–259

Iannario M. and Piccolo D. (2010), A new statistical model for the analysis of customer satisfaction, #' Quality Technology and Quantity management, 7(2) 149–168
Iannario M. and Piccolo D. (2012), CUB models: Statistical methods and empirical evidence, in: Kenett R. S. and Salini S. (eds.), Modern Analysis of Customer Surveys: with applications using R, J. Wiley and Sons, Chichester, 231–258.

Main function for CUBE models without covariates

Description

Estimate and validate a CUBE model without covariates.

Usage

cube000(m, ordinal, starting, maxiter, toler, expinform)

Arguments

Value

An object of the class "CUBE"

References

Iannario, M. (2014). Modelling Uncertainty and Overdispersion in Ordinal Data, Communications in Statistics - Theory and Methods, 43, 771–786
Iannario, M. (2015). Detecting latent components in ordinal data with overdispersion by means of a mixture distribution, Quality & Quantity, 49, 977–987

Main function for CUBE models with covariates

Description

Function to estimate and validate a CUBE model with explicative covariates for all the three parameters.

Usage

cubecov(m, ordinal, Y, W, Z, starting, maxiter, toler)

Arguments

Value

An object of the class "CUBE"

References

Piccolo, D. (2014). Inferential issues on CUBE models with covariates, Communications in Statistics - Theory and Methods, 44, DOI: 10.1080/03610926.2013.821487

Main function for CUBE models with covariates only for feeling

Description

Estimate and validate a CUBE model for ordinal data, with covariates only for explaining the feeling component.

Usage

cubecsi(m, ordinal, W, starting, maxiter, toler)

Arguments

Value

An object of the class "CUBE". For cubecsi, $niter will return a NULL value since the optimization procedure is not iterative but based on "optim" (method = "L-BFGS-B", option hessian=TRUE).
$varmat will return the inverse of the numerically computed Hessian when it is positive definite, otherwise the procedure will return a matrix of NA entries.

See Also

loglikcubecsi, inibestcubecsi, CUBE

Plot an estimated CUBE model

Description

Plotting facility for the CUBE estimation of ordinal responses.

Usage

cubevisual(ordinal,csiplot=FALSE,paiplot=FALSE,...)

Arguments

Details

It represents an estimated CUBE model as a point in the parameter space with the overdispersion being labeled.

Value

For a CUBE model fitted to ordinal, by default it returns a plot of the estimated (1-\pi, 1-\xi) as a point in the parameter space, labeled with the estimated overdispersion \phi. Depending on csiplot and paiplot and on desired output, x and y coordinates may be set to \pi and \xi, respectively.

Examples

data(univer)
ordinal<-univer$global
cubevisual(ordinal,xlim=c(0,0.5),main="Global Satisfaction",
   ylim=c(0.5,1),cex=0.8,digits=3,col="red")

Main function for CUB models with covariates for the uncertainty component

Description

Estimate and validate a CUB model for given ordinal responses, with covariates for explaining the feeling component via a logistic transform.

Usage

cubp0(m, ordinal, Y, maxiter, toler)

Arguments

Value

An object of the class "CUB"

References

Iannario M. and Piccolo D. (2010), A new statistical model for the analysis of customer satisfaction, Quality Technology and Quantity management, 7(2) 149–168
Iannario M. and Piccolo D. (2012). CUB models: Statistical methods and empirical evidence, in: Kenett R. S. and Salini S. (eds.), Modern Analysis of Customer Surveys: with applications using R, J. Wiley and Sons, Chichester, 231–258

Main function for CUB models with covariates for both the uncertainty and the feeling components

Description

Estimate and validate a CUB model for given ordinal responses, with covariates for explaining both the feeling and the uncertainty components by means of logistic transform.

Usage

cubpq(m, ordinal, Y, W, maxiter, toler)

Arguments

Value

An object of the class "CUB"

References

Piccolo D. and D'Elia A. (2008), A new approach for modelling consumers' preferences, Food Quality and Preference, 18, 247–259
Iannario M. and Piccolo D. (2010), A new statistical model for the analysis of customer satisfaction, Quality Technology and Quantitative Management, 17(2) 149–168

See Also

varcovcubpq, loglikcubpq, inibestgama, CUB

Main function for CUB models with a shelter effect

Description

Estimate and validate a CUB model with a shelter effect.

Usage

cubshe(m, ordinal, shelter, maxiter, toler)

Arguments

Value

An object of the class "CUB"

References

Iannario M. (2012). Modelling shelter choices in a class of mixture models for ordinal responses, Statistical Methods and Applications, 21, 1–22

Plot an estimated CUB model with shelter

Description

Plotting facility for the CUB estimation of ordinal responses when a shelter effect is included

Usage

cubshevisual(ordinal,shelter,csiplot=FALSE,paiplot=FALSE,...)

Arguments

Details

It represents an estimated CUB model with shelter effect as a point in the parameter space with shelter estimate indicated as label.

Value

For a CUB model with shelter fitted to ordinal, by default it returns a plot of the estimated (1-\pi, 1-\xi) as a point in the parameter space, labeled with the estimated shelter parameter \delta. Depending on csiplot and paiplot and on desired output, x and y coordinates may be set to \pi and \xi, respectively.

See Also

cubvisual, multicub

Examples

data(univer)
ordinal<-univer$global
cubshevisual(ordinal,shelter=7,digits=3,col="blue",main="Global Satisfaction")

Plot an estimated CUB model

Description

Plotting facility for the CUB estimation of ordinal responses.

Usage

cubvisual(ordinal,csiplot=FALSE,paiplot=FALSE,...)

Arguments

Details

It represents an estimated CUB model as a point in the parameter space with some useful options.

Value

For a CUB model fit to ordinal, by default it returns a plot of the estimated (1-\pi, 1-\xi) as a point in the parameter space. Depending on csiplot and paiplot and on desired output, x and y coordinates may be set to \pi and \xi, respectively.

Examples

data(univer)
ordinal<-univer$global
cubvisual(ordinal,xlim=c(0,0.5),ylim=c(0.5,1),cex=0.8,main="Global Satisfaction")

CUSH model without covariates

Description

Estimate and validate a CUSH model for given ordinal responses, without covariates.

Usage

cush00(m, ordinal, shelter)

Arguments

Value

An object of the class "GEM", "CUSH"

CUSH model with covariates

Description

Estimate and validate a CUSH model for ordinal responses, with covariates to explain the shelter effect.

Usage

cushcov(m, ordinal, X, shelter)

Arguments

Value

An object of the class "GEM", "CUSH"

Mean difference of a discrete random variable

Description

Compute the Gini mean difference of a discrete distribution

Usage

deltaprob(prob)

Arguments

Value

Numeric value of the Gini mean difference of the input probability distribution, computed according to the de Finetti-Paciello formulation.

Examples

prob<-c(0.04,0.04,0.05,0.10,0.21,0.32,0.24)
deltaprob(prob)

Normalized dissimilarity measure

Description

Compute the normalized dissimilarity measure between observed relative frequencies and estimated (theoretical) probabilities of a discrete distribution.

Usage

dissim(proba,probb)

Arguments

Value

Numeric value of the dissimilarity index, assessing the distance to a perfect fit.

Examples

proba<-c(0.01,0.03,0.08,0.07,0.27,0.37,0.17)
probb<-c(0.04,0.04,0.05,0.10,0.21,0.32,0.24)
dissim(proba,probb)

Auxiliary function for the log-likelihood estimation of CUB models

Description

Compute the opposite of the scalar function that is maximized when running the E-M algorithm for CUB models with covariates for the feeling parameter.

Usage

effe01(gama, esterno01, m)

Arguments

Details

It is called as an argument for optim within CUB function for models with covariates for feeling or for both feeling and uncertainty

Auxiliary function for the log-likelihood estimation of CUB models

Description

Compute the opposite of the scalar function that is maximized when running the E-M algorithm for CUB models with covariates for the uncertainty parameter.

Usage

effe10(bet, esterno10)

Arguments

Details

It is called as an argument for optim within CUB function for models with covariates for uncertainty or for both feeling and uncertainty

Auxiliary function for the log-likelihood estimation of CUBE models without covariates

Description

Define the opposite of the scalar function that is maximized when running the E-M algorithm for CUBE models without covariates.

Usage

effecube(paravec, dati, m)

Arguments

Details

It is called as an argument for optim within CUBE function (where no covariate is specified) and "cubeforsim" as the function to minimize.

References

Iannario, M. (2014). Modelling Uncertainty and Overdispersion in Ordinal Data, Communications in Statistics - Theory and Methods, 43, 771–786

Auxiliary function for the log-likelihood estimation of CUBE models with covariates only for the feeling component

Description

Compute the opposite of the scalar function that is maximized when running the E-M algorithm for CUBE models with covariates only for the feeling component.

Usage

effecubecsi(param, ordinal, W, m)

Arguments

Auxiliary function for the log-likelihood estimation of CUSH models with covariates

Description

Compute the opposite of the loglikelihood function for CUSH models with covariates to explain the shelter effect.

Usage

effecush(paravec, esternocush, shelter, m)

Arguments

Details

It is called as an argument for "optim" within CUSH function (when no covariate is included) as the function to minimize.

Auxiliary function for the log-likelihood estimation of IHG models without covariates

Description

Compute the opposite of the log-likelihood function for an IHG model without covariates.

Usage

effeihg(theta, m, freq)

Arguments

Details

It is called as an argument for "optim" within IHG function (when no covariate is specified) as the function to minimize.

Auxiliary function for the log-likelihood estimation of IHG models with covariates

Description

Compute the opposite of the log-likelihood function for an IHG model with covariates for the preference parameter.

Usage

effeihgcov(nu, ordinal, U, m)

Arguments

Details

It is called as an argument for "optim" within IHG function (with covariates) as the function to minimize.

Log-likelihood function of a CUB model without covariates

Description

Compute the log-likelihood function of a CUB model without covariates fitting ordinal responses, possibly with subjects' specific parameters.

Usage

ellecub(m,ordinal,assepai,assecsi)

Arguments

See Also

loglikCUB

Examples

m<-7
n0<-230
n1<-270
bet<-c(-1.5,1.2)
gama<-c(0.5,-1.2)
pai0<-logis(0,bet); csi0<-logis(0,gama)
pai1<-logis(1,bet); csi1<-logis(1,gama)
ordinal0<-simcub(n0,m,pai0,csi0)
ordinal1<-simcub(n1,m,pai1,csi1)
ordinal<-c(ordinal0,ordinal1)
assepai<-c(rep(pai0,n0),rep(pai1,n1))
assecsi<-c(rep(csi0,n0),rep(csi1,n1))
lli<-ellecub(m,ordinal,assepai,assecsi)

Log-likelihood function for gecub distribution

Description

Log-likelihood function for gecub distribution

Usage

ellegecub(ordinal,Y,W,X,bet,gama,omega,shelter)

Arguments

Expectation of CUB distributions

Description

Compute the expectation of a CUB model without covariates.

Usage

expcub00(m,pai,csi)

Arguments

References

Piccolo D. (2003). On the moments of a mixture of uniform and shifted binomial random variables. Quaderni di Statistica, 5, 85–104

See Also

varcub00, expcube, varcube

Examples

m<-10
pai<-0.3
csi<-0.7
meancub<-expcub00(m,pai,csi)

Expectation of CUBE models

Description

Compute the expectation of a CUBE model without covariates.

Usage

expcube(m,pai,csi,phi)

Arguments

References

Iannario M. (2014). Modelling Uncertainty and Overdispersion in Ordinal Data, Communications in Statistics - Theory and Methods, 43, 771–786
Iannario, M. (2015). Detecting latent components in ordinal data with overdispersion by means of a mixture distribution, Quality & Quantity, 49, 977–987

See Also

varcube, varcub00, expcub00

Examples

m<-10
pai<-0.1
csi<-0.7
phi<-0.2
meancube<-expcube(m,pai,csi,phi)

S3 method "fitted" for class "GEM"

Description

S3 method fitted for objects of class GEM.

Usage

## S3 method for class 'GEM'
fitted(object, ...)

Arguments

Details

Returns the fitted probability distribution for GEM models with no covariates. If only one dichotomous covariate is included in the model to explain some components, it returns the fitted probability distribution for each profile.

See Also

GEM

Examples

fitcub<-GEM(Formula(global~0|freqserv|0),family="cub",data=univer)
fitted(fitcub,digits=4)

Main function for CUB models with covariates for all the components

Description

Function to estimate and validate a CUB model for given ordinal responses, with covariates for explaining all the components and the shelter effect.

Usage

gecubpqs(ordinal,Y,W,X,shelter,theta0,maxiter,toler)

Arguments

Value

An object of the class "CUB"

References

Piccolo D. and D'Elia A. (2008), A new approach for modelling consumers' preferences, Food Quality and Preference, 18, 247–259

Iannario M. and Piccolo D. (2010), A new statistical model for the analysis of customer satisfaction, #' Quality Technology and Quantity management, 7(2) 149–168
Iannario M. and Piccolo D. (2012), CUB models: Statistical methods and empirical evidence, in: Kenett R. S. and Salini S. (eds.), Modern Analysis of Customer Surveys: with applications using R, J. Wiley and Sons, Chichester, 231–258.

Normalized Gini heterogeneity index

Description

Compute the normalized Gini heterogeneity index for a given discrete probability distribution.

Usage

gini(prob)

Arguments

See Also

laakso

Examples

prob<-c(0.04,0.04,0.05,0.10,0.21,0.32,0.24)
gini(prob)

Main function for IHG models without covariates

Description

Estimate and validate an IHG model without covariates for given ordinal responses.

Usage

ihg00(m, ordinal)

Arguments

Details

The optimization procedure is run via "optim", option method="Brent" for constrained optimization (lower bound = 0, upper bound=1).

Value

An object of the class "IHG"

An object of the class "IHG"

Main function for IHG models with covariates

Description

Estimate and validate an IHG model for given ordinal responses, with covariates to explain the preference parameter.

Usage

ihgcov(m, ordinal, U)

Arguments

Details

The optimization procedure is run via "optim", option method="Brent" for constrained optimization (lower bound = 0, upper bound=1).

Value

An object of the class "IHG"

An object of the class "IHG"

Preliminary estimators for CUB models without covariates

Description

Compute preliminary parameter estimates of a CUB model without covariates for given ordinal responses. These preliminary estimators are used within the package code to start the E-M algorithm.

Usage

inibest(m,freq)

Arguments

Value

A vector (\pi,\xi) of the initial parameter estimates for a CUB model without covariates, given the absolute frequency distribution of ordinal responses

References

Iannario M. (2009). A comparison of preliminary estimators in a class of ordinal data models, Statistica & Applicazioni, VII, 25–44
Iannario M. (2012). Preliminary estimators for a mixture model of ordinal data, Advances in Data Analysis and Classification, 6, 163–184

See Also

inibestgama

Examples

m<-9
freq<-c(10,24,28,36,50,43,23,12,5)
estim<-inibest(m,freq) 
pai<-estim[1]
csi<-estim[2]

Naive estimates for CUBE models without covariates

Description

Compute naive parameter estimates of a CUBE model without covariates for given ordinal responses. These preliminary estimators are used within the package code to start the E-M algorithm.

Usage

inibestcube(m,ordinal)

Arguments

Value

A vector (\pi, \xi ,\phi) of parameter estimates of a CUBE model without covariates.

See Also

inibestcubecov, inibestcubecsi

Examples

data(relgoods)
m<-10
ordinal<-relgoods$SocialNetwork
estim<-inibestcube(m,ordinal)     # Preliminary estimates (pai,csi,phi)

Preliminary parameter estimates for CUBE models with covariates

Description

Compute preliminary parameter estimates for a CUBE model with covariates for all the three parameters. These estimates are set as initial values to start the E-M algorithm within maximum likelihood estimation.

Usage

inibestcubecov(m,ordinal,Y,W,Z)

Arguments

Value

A vector (inibet, inigama, inialpha) of preliminary estimates of parameter vectors for \pi = \pi(\bold{\beta}), \xi=\xi(\bold{\gamma}), \phi=\phi(\bold{\alpha}), respectively, of a CUBE model with covariates for all the three parameters. In details, inibet, inigama and inialpha have length equal to NCOL(Y)+1, NCOL(W)+1 and NCOL(Z)+1, respectively, to account for an intercept term for each component.

See Also

inibestcube, inibestcubecsi, inibestgama

Examples

data(relgoods)
m<-10 
naord<-which(is.na(relgoods$Tv))
nacovpai<-which(is.na(relgoods$Gender))
nacovcsi<-which(is.na(relgoods$year.12))
nacovphi<-which(is.na(relgoods$EducationDegree))
na<-union(union(naord,nacovpai),union(nacovcsi,nacovphi))
ordinal<-relgoods$Tv[-na]
Y<-relgoods$Gender[-na]
W<-relgoods$year.12[-na]
Z<-relgoods$EducationDegree[-na]
ini<-inibestcubecov(m,ordinal,Y,W,Z)
p<-NCOL(Y)
q<-NCOL(W)
inibet<-ini[1:(p+1)]               # Preliminary estimates for uncertainty 
inigama<-ini[(p+2):(p+q+2)]        # Preliminary estimates for feeling 
inialpha<-ini[(p+q+3):length(ini)] # Preliminary estimates for overdispersion

Preliminary estimates of parameters for CUBE models with covariates only for feeling

Description

Compute preliminary parameter estimates of a CUBE model with covariates only for feeling, given ordinal responses. These estimates are set as initial values to start the corresponding E-M algorithm within the package.

Usage

inibestcubecsi(m,ordinal,W,starting,maxiter,toler)

Arguments

Details

Preliminary estimates for the uncertainty and the overdispersion parameters are computed by short runs of EM. As to the feeling component, it considers the nested CUB model with covariates and calls inibestgama to derive initial estimates for the coefficients of the selected covariates for feeling.

Value

A vector (pai, gamaest, phi), where pai is the initial estimate for the uncertainty parameter, gamaest is the vector of initial estimates for the feeling component (including an intercept term in the first entry), and phi is the initial estimate for the overdispersion parameter.

See Also

inibestcube, inibestcubecov, inibestgama

Examples

data(relgoods)
isnacov<-which(is.na(relgoods$Gender))
isnaord<-which(is.na(relgoods$Tv))
na<-union(isnacov,isnaord)
ordinal<-relgoods$Tv[-na]; W<-relgoods$Gender[-na]
m<-10
starting<-rep(0.1,3)
ini<-inibestcubecsi(m,ordinal,W,starting,maxiter=100,toler=1e-3)
nparam<-length(ini)
pai<-ini[1]                 # Preliminary estimates for uncertainty component
gamaest<-ini[2:(nparam-1)]  # Preliminary estimates for coefficients of feeling covariates
phi<-ini[nparam]            # Preliminary estimates for overdispersion component

Preliminary parameter estimates of a CUB model with covariates for feeling

Description

Compute preliminary parameter estimates for the feeling component of a CUB model fitted to ordinal responses These estimates are set as initial values for parameters to start the E-M algorithm.

Usage

inibestgama(m,ordinal,W)

Arguments

Value

A vector of length equal to NCOL(W)+1, whose entries are the preliminary estimates of the parameters for the feeling component, including an intercept term as first entry.

References

Iannario M. (2008). Selecting feeling covariates in rating surveys, Rivista di Statistica Applicata, 20, 103–116
Iannario M. (2009). A comparison of preliminary estimators in a class of ordinal data models, Statistica & Applicazioni, VII, 25–44
Iannario M. (2012). Preliminary estimators for a mixture model of ordinal data, Advances in Data Analysis and Classification, 6, 163–184

See Also

inibest, inibestcubecsi

Examples

data(univer)
m<-7; ordinal<-univer$global; cov<-univer$diploma
ini<-inibestgama(m,ordinal,W=cov)

Grid-based preliminary parameter estimates for CUB models

Description

Compute the log-likelihood function of a CUB model with parameter vector (\pi, \xi) ranging in the Cartesian product between x and y, for a given absolute frequency distribution.

Usage

inigrid(m,freq,x,y)

Arguments

Value

It returns the parameter vector corresponding to the maximum value of the log-likelihood for a CUB model without covariates for given frequencies.

See Also

inibest

Examples

m<-9
x<-c(0.1,0.4,0.6,0.8)
y<-c(0.2, 0.5,0.7)
freq<-c(10,24,28,36,50,43,23,12,5)
ini<-inigrid(m,freq,x,y)
pai<-ini[1]
csi<-ini[2]

Moment estimate for the preference parameter of the IHG distribution

Description

Compute the moment estimate of the preference parameter of the IHG distribution. This preliminary estimate is set as initial value within the optimization procedure for an IHG model fitting the observed frequencies.

Usage

iniihg(m,freq)

Arguments

Value

Moment estimator of the preference parameter \theta.

References

D'Elia A. (2003). Modelling ranks using the inverse hypergeometric distribution, Statistical Modelling: an International Journal, 3, 65–78.

See Also

inibest, inibestcube

Examples

m<-9
freq<-c(70,51,48,38,29,23,12,10,5)
initheta<-iniihg(m,freq)

Sequence of combinatorial coefficients

Description

Compute the sequence of binomial coefficients {m-1}\choose{r-1}, for r= 1, \dots, m, and then returns a vector of the same length as ordinal, whose i-th component is the corresponding binomial coefficient {m-1}\choose{r_i-1}

Usage

kkk(m, ordinal)

Arguments

Normalized Laakso and Taagepera heterogeneity index

Description

Compute the normalized Laakso and Taagepera heterogeneity index for a given discrete probability distribution.

Usage

laakso(prob)

Arguments

References

Laakso, M. and Taagepera, R. (1989). Effective number of parties: a measure with application to West Europe, Comparative Political Studies, 12, 3–27.

See Also

gini

Examples

prob<-c(0.04,0.04,0.05,0.10,0.21,0.32,0.24)
laakso(prob)

logLik S3 Method for class "GEM"

Description

S3 method: logLik() for objects of class "GEM".

Usage

## S3 method for class 'GEM'
logLik(object, ...)

Arguments

Value

Log-likelihood at the final ML estimates for parameters of the fitted GEM model.

See Also

loglikCUB, loglikCUBE, GEM, loglikIHG, loglikCUSH, BIC

The logistic transform

Description

Create a matrix YY binding array Y with a vector of ones, placed as the first column of YY. It applies the logistic transform componentwise to the standard matrix multiplication between YY and param.

Usage

logis(Y,param)

Arguments

Value

Return a vector whose length is NROW(Y) and whose i-th component is the logistic function at the scalar product between the i-th row of YY and the vector param.

Examples

n<-50 
Y<-sample(c(1,2,3),n,replace=TRUE) 
param<-c(0.2,0.7)
logis(Y,param)

Log-likelihood function for CUB models

Description

Compute the log-likelihood value of a CUB model fitting given data, with or without covariates to explain the feeling and uncertainty components, or for extended CUB models with shelter effect.

Usage

loglikCUB(ordinal,m,param,Y=0,W=0,X=0,shelter=0)

Arguments

Details

If no covariate is included in the model, then param should be given in the form (\pi,\xi). More generally, it should have the form (\bold{\beta,\gamma)} where, respectively, \bold{\beta} and \bold{\gamma} are the vectors of coefficients explaining the uncertainty and the feeling components, with length NCOL(Y)+1 and NCOL(W)+1 to account for an intercept term in the first entry. When shelter effect is considered, param corresponds to the first possibile parameterization and hence should be given as (pai1,pai2,csi). No missing value should be present neither for ordinal nor for covariate matrices: thus, deletion or imputation procedures should be preliminarily run.

See Also

logLik

Examples

## Log-likelihood of a CUB model with no covariate
m<-9; n<-300
pai<-0.6; csi<-0.4
ordinal<-simcub(n,m,pai,csi)
param<-c(pai,csi)
loglikcub<-loglikCUB(ordinal,m,param)
##################################
## Log-likelihood of a CUB model with covariate for uncertainty
data(relgoods)
m<-10
naord<-which(is.na(relgoods$Physician))
nacov<-which(is.na(relgoods$Gender))
na<-union(naord,nacov)
ordinal<-relgoods$Physician[-na]; Y<-relgoods$Gender[-na]
bbet<-c(-0.81,0.93); ccsi<-0.2
param<-c(bbet,ccsi)
loglikcubp0<-loglikCUB(ordinal,m,param,Y=Y)
#######################
## Log-likelihood of a CUB model with covariate for feeling
data(relgoods)
m<-10
naord<-which(is.na(relgoods$Physician))
nacov<-which(is.na(relgoods$Gender))
na<-union(naord,nacov)
ordinal<-relgoods$Physician[-na]; W<-relgoods$Gender[-na]
pai<-0.44; gama<-c(-0.91,-0.7)
param<-c(pai,gama)
loglikcub0q<-loglikCUB(ordinal,m,param,W=W)
#######################
## Log-likelihood of a CUB model with covariates for both parameters
data(relgoods)
m<-10
naord<-which(is.na(relgoods$Walking))
nacovpai<-which(is.na(relgoods$Gender))
nacovcsi<-which(is.na(relgoods$Smoking))
na<-union(naord,union(nacovpai,nacovcsi))
ordinal<-relgoods$Walking[-na]
Y<-relgoods$Gender[-na]; W<-relgoods$Smoking[-na]
bet<-c(-0.45,-0.48); gama<-c(-0.55,-0.43)
param<-c(bet,gama)
loglikcubpq<-loglikCUB(ordinal,m,param,Y=Y,W=W)
#######################
### Log-likelihood of a CUB model with shelter effect
m<-7; n<-400
pai<-0.7; csi<-0.16; delta<-0.15
shelter<-5
ordinal<-simcubshe(n,m,pai,csi,delta,shelter)
pai1<- pai*(1-delta); pai2<-1-pai1-delta
param<-c(pai1,pai2,csi)
loglik<-loglikCUB(ordinal,m,param,shelter=shelter)

Log-likelihood function for CUBE models

Description

Compute the log-likelihood function for CUBE models. It is possible to include covariates in the model for explaining the feeling component or all the three parameters.

Usage

loglikCUBE(ordinal,m,param,Y=0,W=0,Z=0)

Arguments

Details

If no covariate is included in the model, then param has the form (\pi,\xi,\phi). More generally, it has the form (\bold{\beta,\gamma,\alpha)} where, respectively, \bold{\beta},\bold{\gamma}, \bold{\alpha} are the vectors of coefficients explaining the uncertainty, the feeling and the overdispersion components, with length NCOL(Y)+1, NCOL(W)+1, NCOL(Z)+1 to account for an intercept term in the first entry. No missing value should be present neither for ordinal nor for covariate matrices: thus, deletion or imputation procedures should be preliminarily run.

See Also

logLik

Examples

#### Log-likelihood of a CUBE model with no covariate
m<-7; n<-400
pai<-0.83; csi<-0.19; phi<-0.045
ordinal<-simcube(n,m,pai,csi,phi)
loglik<-loglikCUBE(ordinal,m,param=c(pai,csi,phi))
##################################
#### Log-likelihood of a CUBE model with covariate for feeling
data(relgoods)
m<-10
nacov<-which(is.na(relgoods$BirthYear))
naord<-which(is.na(relgoods$Tv))
na<-union(nacov,naord)
age<-2014-relgoods$BirthYear[-na]
lage<-log(age)-mean(log(age))
ordinal<-relgoods$Tv[-na]; W<-lage
pai<-0.63; gama<-c(-0.61,-0.31); phi<-0.16
param<-c(pai,gama,phi)
loglik<-loglikCUBE(ordinal,m,param,W=W)
########## Log-likelihood of a CUBE model with covariates for all parameters
Y<-W<-Z<-lage
bet<-c(0.18, 1.03); gama<-c(-0.6, -0.3); alpha<-c(-2.3,0.92)
param<-c(bet,gama,alpha)
loglik<-loglikCUBE(ordinal,m,param,Y=Y,W=W,Z=Z)

Log-likelihood function for CUSH models

Description

Compute the log-likelihood function for CUSH models with or without covariates to explain the shelter effect.

Usage

loglikCUSH(ordinal,m,param,shelter,X=0)

Arguments

Details

If no covariate is included in the model, then param is the estimate of the shelter parameter (delta), otherwise param has length equal to NCOL(X) + 1 to account for an intercept term (first entry). No missing value should be present neither for ordinal nor for X.

See Also

GEM, logLik

Examples

## Log-likelihood of CUSH model without covariates
n<-300
m<-7
shelter<-2; delta<-0.4
ordinal<-simcush(n,m,delta,shelter)
loglik<-loglikCUSH(ordinal,m,param=delta,shelter)
#####################
## Log-likelihood of CUSH model with covariates
data(relgoods)
m<-10
naord<-which(is.na(relgoods$SocialNetwork))
nacov<-which(is.na(relgoods$Gender))
na<-union(nacov,naord)
ordinal<-relgoods$SocialNetwork[-na]; cov<-relgoods$Gender[-na]
omega<-c(-2.29, 0.62)
loglikcov<-loglikCUSH(ordinal,m,param=omega,shelter=1,X=cov)

Log-likelihood function for IHG models

Description

Compute the log-likelihood function for IHG models with or without covariates to explain the preference parameter.

Usage

loglikIHG(ordinal,m,param,U=0)

Arguments

Details

If no covariate is included in the model, then param is the estimate of the preference parameter (theta), otherwise param has length equal to NCOL(U) + 1 to account for an intercept term (first entry). No missing value should be present neither for ordinal nor for U.

See Also

GEM, logLik

Examples

#### Log-likelihood of an IHG model with no covariate
m<-10; theta<-0.14; n<-300
ordinal<-simihg(n,m,theta)
loglik<-loglikIHG(ordinal,m,param=theta)
##################################
#### Log-likelihood of a IHG model with covariate 
data(relgoods)
m<-10
naord<-which(is.na(relgoods$HandWork))
nacov<-which(is.na(relgoods$Gender))
na<-union(naord,nacov)
ordinal<-relgoods$HandWork[-na]; U<-relgoods$Gender[-na]
nu<-c(-1.55,-0.11)     # first entry: intercept term
loglik<-loglikIHG(ordinal,m,param=nu,U=U); loglik

Log-likelihood function of a CUB model without covariates

Description

Compute the log-likelihood function of a CUB model without covariates for a given absolute frequency distribution.

Usage

loglikcub00(m, freq, pai, csi)

Arguments

Log-likelihood function of a CUB model with covariates for the feeling component

Description

Compute the log-likelihood function of a CUB model fitting ordinal data, with q covariates for explaining the feeling component.

Usage

loglikcub0q(m, ordinal, W, pai, gama)

Arguments

Log-likelihood function of a CUBE model with covariates

Description

Compute the log-likelihood function of a CUBE model for ordinal responses, with covariates for explaining all the three parameters.

Usage

loglikcubecov(m, ordinal, Y, W, Z, bet, gama, alpha)

Arguments

Log-likelihood function of CUBE model with covariates only for feeling

Description

Compute the log-likelihood function of a CUBE model for ordinal data with subjects' covariates only for feeling.

Usage

loglikcubecsi(m, ordinal, W, pai, gama, phi)

Arguments

See Also

internal

Log-likelihood function of CUBE models for ordinal data

Description

Compute the log-likelihood function of a CUBE model without covariates for ordinal responses, possibly with different vectors of parameters for each observation.

Usage

loglikcuben(m, ordinal, assepai, assecsi, assephi)

Arguments

See Also

loglikCUBE

Examples

m<-8
n0<-230;  n1<-270
bet<-c(-1.5,1.2)
gama<-c(0.5,-1.2)
alpha<-c(-1.2,-0.5)
pai0<-1/(1+exp(-bet[1])); csi0<-1/(1+exp(-gama[1])); phi0<-exp(alpha[1])
ordinal0<-simcube(n0,m,pai0,csi0,phi0)
pai1<-1/(1+exp(-sum(bet))); csi1<-1/(1+exp(-sum(gama))); phi1<-exp(sum(alpha))
ordinal1<-simcube(n1,m,pai1,csi1,phi1)
ordinal<-c(ordinal0,ordinal1)
assepai<-c(rep(pai0,n0),rep(pai1,n1))
assecsi<-c(rep(csi0,n0),rep(csi1,n1))
assephi<-c(rep(phi0,n0),rep(phi1,n1))
lli<-loglikcuben(m,ordinal,assepai,assecsi,assephi)

Log-likelihood function of a CUB model with covariates for the uncertainty component

Description

Compute the log-likelihood function of a CUB model fitting ordinal responses with covariates for explaining the uncertainty component.

Usage

loglikcubp0(m, ordinal, Y, bbet, ccsi)

Arguments

Log-likelihood function of a CUB model with covariates for both feeling and uncertainty

Description

Compute the log-likelihood function of a CUB model fitting ordinal data with covariates for explaining both the feeling and the uncertainty components.

Usage

loglikcubpq(m, ordinal, Y, W, bet, gama)

Arguments

Log-likelihood of a CUB model with shelter effect

Description

Compute the log-likelihood of a CUB model with a shelter effect for the given absolute frequency distribution.

Usage

loglikcubshe(m, freq, pai1, pai2, csi, shelter)

Arguments

Log-likelihood function for a CUSH model without covariates

Description

Compute the log-likelihood function for a CUSH model without covariate for the given ordinal responses.

Usage

loglikcush00(m,ordinal,delta,shelter)

Arguments

See Also

GEM

Log-likelihood function for a CUSH model with covariates

Description

Compute the log-likelihood function for a CUSH model with covariates for the given ordinal responses.

Usage

loglikcushcov(m, ordinal, X, omega, shelter)

Arguments

Log-likelihood function for the IHG model with covariates

Description

Compute the log-likelihood function for the IHG model with covariates to explain the preference parameter.

Usage

loglikihgcov(m, ordinal, U, nu)

Arguments

See Also

loglikIHG

Logarithmic score

Description

Compute the logarithmic score of a CUB model with covariates both for the uncertainty and the feeling parameters.

Usage

logscore(m,ordinal,Y,W,bet,gama)

Arguments

Details

No missing value should be present neither for ordinal nor for covariate matrices: thus, deletion or imputation procedures should be preliminarily run.

References

Tutz, G. (2012). Regression for Categorical Data, Cambridge University Press, Cambridge

Examples

data(relgoods)
m<-10
naord<-which(is.na(relgoods$Walking))
nacovpai<-which(is.na(relgoods$Gender))
nacovcsi<-which(is.na(relgoods$Smoking))
na<-union(naord,union(nacovpai,nacovcsi))
ordinal<-relgoods$Walking[-na]
Y<-relgoods$Gender[-na]
W<-relgoods$Smoking[-na]
bet<-c(-0.45,-0.48)
gama<-c(-0.55,-0.43)
logscore(m,ordinal,Y=Y,W=W,bet,gama)

Plot facilities for GEM objects

Description

Plot facilities for objects of class "GEM".

Usage

makeplot(object)

Arguments

Details

Returns a plot comparing fitted probabilities and observed relative frequencies for GEM models without covariates. If only one explanatory dichotomous variable is included in the model for one or all components, then the function returns a plot comparing the distributions of the responses conditioned to the value of the covariate.

See Also

cubvisual, cubevisual, cubshevisual, multicub, multicube

Joint plot of estimated CUB models in the parameter space

Description

Return a plot of estimated CUB models represented as points in the parameter space.

Usage

multicub(listord,mvett,csiplot=FALSE,paiplot=FALSE,...)

Arguments

Value

Fit a CUB model to list elements, and then by default it returns a plot of the estimated (1-\pi, 1-\xi) as points in the parameter space. Depending on csiplot and paiplot and on desired output, x and y coordinates may be set to \pi and \xi, respectively.

Examples

data(univer)
listord<-univer[,8:12]
multicub(listord,colours=rep("red",5),cex=c(0.4,0.6,0.8,1,1.2),
  pch=c(1,2,3,4,5),xlim=c(0,0.4),ylim=c(0.75,1),pos=c(1,3,3,3,3))
###############################
m1<-5; m2<-7;  m3<-9
pai<-0.7;csi<-0.6
n1<-1000; n2<-500; n3<-1500
ord1<-simcub(n1,m1,pai,csi)
ord2<-simcub(n2,m2,pai,csi)
ord3<-simcub(n3,m3,pai,csi)
listord<-list(ord1,ord2,ord3)
multicub(listord,labels=c("m=5","m=7","m=9"),pos=c(3,1,4))

Joint plot of estimated CUBE models in the parameter space

Description

Return a plot of estimated CUBE models represented as points in the parameter space, where the overdispersion is labeled.

Usage

multicube(listord,mvett,csiplot=FALSE,paiplot=FALSE,...)

Arguments

Value

Fit a CUBE model to list elements, and then by default it returns a plot of the estimated (1-\pi, 1-\xi) as points in the parameter space, labeled with the estimated overdispersion. Depending on csiplot and paiplot and on desired output, x and y coordinates may be set to \pi and \xi, respectively.

Examples

m1<-5; m2<-7;  m3<-9
pai<-0.7;csi<-0.6;phi=0.1
n1<-1000; n2<-500; n3<-1500
ord1<-simcube(n1,m1,pai,csi,phi)
ord2<-simcube(n2,m2,pai,csi,phi)
ord3<-simcube(n3,m3,pai,csi,phi)
listord<-list(ord1,ord2,ord3)
multicube(listord,labels=c("m=5","m=7","m=9"),pos=c(3,1,4),expinform=TRUE)

Generic function for coefficient names

Description

Generic function for names of parameter estimates of object of class "GEM".

Usage

parnames(object)

Arguments

See Also

summary

Examples

data(univer);attach(univer)
model<-GEM(Formula(officeho~0|0|0),family="cub",shelter=7)
model 

Plot of the log-likelihood function of the IHG distribution

Description

Plot the log-likelihood function of an IHG model fitted to a given absolute frequency distribution, over the whole support of the preference parameter. It returns also the ML estimate.

Usage

plotloglikihg(m,freq)

Arguments

See Also

loglikIHG

Examples

m<-7
freq<-c(828,275,202,178,143,110,101)
max<-plotloglikihg(m,freq)

S3 method: print for class "GEM"

Description

S3 method print for objects of class GEM.

Usage

## S3 method for class 'GEM'
print(x, ...)

Arguments

Value

Brief summary results of the fitting procedure, including parameter estimates, their standard errors and the executed call.

Probability distribution of a shifted Binomial random variable

Description

Return the shifted Binomial probability distribution.

Usage

probbit(m,csi)

Arguments

Value

The vector of the probability distribution of a shifted Binomial model.

See Also

bitcsi, probcub00

Examples

m<-7
csi<-0.7
pr<-probbit(m,csi)
plot(1:m,pr,type="h",main="Shifted Binomial probability distribution",xlab="Categories")
points(1:m,pr,pch=19)

Probability distribution of a CUB model without covariates

Description

Compute the probability distribution of a CUB model without covariates.

Usage

probcub00(m,pai,csi)

Arguments

Value

The vector of the probability distribution of a CUB model.

References

Piccolo D. (2003). On the moments of a mixture of uniform and shifted binomial random variables. Quaderni di Statistica, 5, 85–104

See Also

bitcsi, probcub0q, probcubp0, probcubpq

Examples

m<-9
pai<-0.3
csi<-0.8
pr<-probcub00(m,pai,csi)
plot(1:m,pr,type="h",main="CUB probability distribution",xlab="Ordinal categories")
points(1:m,pr,pch=19)

Probability distribution of a CUB model with covariates for the feeling component

Description

Compute the probability distribution of a CUB model with covariates for the feeling component.

Usage

probcub0q(m,ordinal,W,pai,gama)

Arguments

Value

A vector of the same length as ordinal, whose i-th component is the probability of the i-th observation according to a CUB distribution with the corresponding values of the covariates for the feeling component and coefficients specified in gama.

References

Piccolo D. (2006). Observed Information Matrix for MUB Models, Quaderni di Statistica, 8, 33–78
Piccolo D. and D'Elia A. (2008). A new approach for modelling consumers' preferences, Food Quality and Preference, 18, 247–259
Iannario M. and Piccolo D. (2012). CUB models: Statistical methods and empirical evidence, in: Kenett R. S. and Salini S. (eds.), Modern Analysis of Customer Surveys: with applications using R, J. Wiley and Sons, Chichester, 231–258

See Also

bitgama, probcub00, probcubp0, probcubpq

Examples

data(relgoods)
m<-10
naord<-which(is.na(relgoods$Physician))
nacov<-which(is.na(relgoods$Gender))
na<-union(naord,nacov)
ordinal<-relgoods$Physician[-na]
W<-relgoods$Gender[-na]
pai<-0.44; gama<-c(-0.91,-0.7)
pr<-probcub0q(m,ordinal,W,pai,gama)

Probability distribution of a CUBE model without covariates

Description

Compute the probability distribution of a CUBE model without covariates.

Usage

probcube(m,pai,csi,phi)

Arguments

Value

The vector of the probability distribution of a CUBE model without covariates.

References

Iannario, M. (2014). Modelling Uncertainty and Overdispersion in Ordinal Data, Communications in Statistics - Theory and Methods, 43, 771–786

See Also

betar, betabinomial

Examples

m<-9
pai<-0.3
csi<-0.8
phi<-0.1
pr<-probcube(m,pai,csi,phi)
plot(1:m,pr,type="h", main="CUBE probability distribution",xlab="Ordinal categories")
points(1:m,pr,pch=19)

Probability distribution of a CUB model with covariates for the uncertainty component

Description

Compute the probability distribution of a CUB model with covariates for the uncertainty component.

Usage

probcubp0(m,ordinal,Y,bet,csi)

Arguments

Value

A vector of the same length as ordinal, whose i-th component is the probability of the i-th observation according to a CUB model with the corresponding values of the covariates for the uncertainty component and coefficients for the covariates specified in bet.

References

Piccolo D. (2006). Observed Information Matrix for MUB Models, Quaderni di Statistica, 8, 33–78
Piccolo D. and D'Elia A. (2008). A new approach for modelling consumers' preferences, Food Quality and Preference, 18, 247–259
Iannario M. and Piccolo D. (2012). CUB models: Statistical methods and empirical evidence, in: Kenett R. S. and Salini S. (eds.), Modern Analysis of Customer Surveys: with applications using R, J. Wiley and Sons, Chichester, 231–258

See Also

bitgama, probcub00, probcubpq, probcub0q

Examples

data(relgoods)
m<-10
naord<-which(is.na(relgoods$Physician))
nacov<-which(is.na(relgoods$Gender))
na<-union(naord,nacov)
ordinal<-relgoods$Physician[-na]
Y<-relgoods$Gender[-na]
bet<-c(-0.81,0.93); csi<-0.20
probi<-probcubp0(m,ordinal,Y,bet,csi)

Probability distribution of a CUB model with covariates for both feeling and uncertainty

Description

Compute the probability distribution of a CUB model with covariates for both the feeling and the uncertainty components.

Usage

probcubpq(m,ordinal,Y,W,bet,gama)

Arguments

Value

A vector of the same length as ordinal, whose i-th component is the probability of the i-th rating according to a CUB distribution with given covariates for both uncertainty and feeling, and specified coefficients vectors bet and gama, respectively.

References

Piccolo D. (2006). Observed Information Matrix for MUB Models, Quaderni di Statistica, 8, 33–78
Piccolo D. and D'Elia A. (2008). A new approach for modelling consumers' preferences, Food Quality and Preference, 18, 247–259
Iannario M. and Piccolo D. (2012). CUB models: Statistical methods and empirical evidence, in: Kenett R. S. and Salini S. (eds.), Modern Analysis of Customer Surveys: with applications using R, J. Wiley and Sons, Chichester, 231–258

See Also

bitgama, probcub00, probcubp0, probcub0q

Examples

data(relgoods)
m<-10
naord<-which(is.na(relgoods$Physician))
nacov<-which(is.na(relgoods$Gender))
na<-union(naord,nacov)
ordinal<-relgoods$Physician[-na]
W<-Y<-relgoods$Gender[-na]
gama<-c(-0.91,-0.7); bet<-c(-0.81,0.93)
probi<-probcubpq(m,ordinal,Y,W,bet,gama)

probcubshe1

Description

Probability distribution of an extended CUB model with a shelter effect.

Usage

probcubshe1(m,pai1,pai2,csi,shelter)

Arguments

Details

An extended CUB model is a mixture of three components: a shifted Binomial distribution with probability of success \xi, a discrete uniform distribution with support \{1,...,m\}, and a degenerate distribution with unit mass at the shelter category (shelter).

Value

The vector of the probability distribution of an extended CUB model with a shelter effect at the shelter category

References

Iannario M. (2012). Modelling shelter choices in a class of mixture models for ordinal responses, Statistical Methods and Applications, 21, 1–22

See Also

probcubshe2, probcubshe3

Examples

m<-8
pai1<-0.5
pai2<-0.3
csi<-0.4
shelter<-6
pr<-probcubshe1(m,pai1,pai2,csi,shelter)
plot(1:m,pr,type="h",main="Extended CUB probability distribution with shelter effect",
xlab="Ordinal categories")
points(1:m,pr,pch=19)

probcubshe2

Description

Probability distribution of a CUB model with explicit shelter effect

Usage

probcubshe2(m,pai,csi,delta,shelter)

Arguments

Details

A CUB model with explicit shelter effect is a mixture of two components: a CUB distribution with uncertainty parameter \pi and feeling parameter \xi, and a degenerate distribution with unit mass at the shelter category (shelter) with mixing coefficient specified by \delta.

Value

The vector of the probability distribution of a CUB model with explicit shelter effect.

References

Iannario M. (2012). Modelling shelter choices in a class of mixture models for ordinal responses, Statistical Methods and Applications, 21, 1–22

See Also

probcubshe1, probcubshe3

Examples

m<-8
pai1<-0.5
pai2<-0.3
csi<-0.4
shelter<-6
delta<-1-pai1-pai2
pai<-pai1/(1-delta)
pr2<-probcubshe2(m,pai,csi,delta,shelter)
plot(1:m,pr2,type="h", main="CUB probability distribution with 
explicit shelter effect",xlab="Ordinal categories")
points(1:m,pr2,pch=19)

probcubshe3

Description

Probability distribution of a CUB model with explicit shelter effect: satisficing interpretation

Usage

probcubshe3(m,lambda,eta,csi,shelter)

Arguments

Details

The "satisficing interpretation" provides a parametrization for CUB models with explicit shelter effect as a mixture of two components: a shifted Binomial distribution with feeling parameter \xi (meditated choice), and a mixture of a degenerate distribution with unit mass at the shelter category (shelter) and a discrete uniform distribution over m categories, with mixing coefficient specified by \eta (lazy selection of a category).

Value

The vector of the probability distribution of a CUB model with shelter effect.

References

Iannario M. (2012). Modelling shelter choices in a class of mixture models for ordinal responses, Statistical Methods and Applications, 21, 1–22

See Also

probcubshe1, probcubshe2

Examples

m<-8
pai1<-0.5
pai2<-0.3
csi<-0.4
shelter<-6
lambda<-pai1
eta<-1-pai2/(1-pai1)
pr3<-probcubshe3(m,lambda,eta,csi,shelter)
plot(1:m,pr3,type="h",main="CUB probability distribution with explicit 
shelter effect",xlab="Ordinal categories")
points(1:m,pr3,pch=19)

Probability distribution of a CUSH model

Description

Compute the probability distribution of a CUSH model without covariates, that is a mixture of a degenerate random variable with mass at the shelter category and the Uniform distribution.

Usage

probcush(m,delta,shelter)

Arguments

Value

The vector of the probability distribution of a CUSH model without covariates.

References

Capecchi S. and Piccolo D. (2017). Dealing with heterogeneity in ordinal responses, Quality and Quantity, 51(5), 2375–2393
Capecchi S. and Iannario M. (2016). Gini heterogeneity index for detecting uncertainty in ordinal data surveys, Metron, 74(2), 223–232

Examples

m<-10
shelter<-1
delta<-0.4
pr<-probcush(m,delta,shelter)
plot(1:m,pr,type="h",xlab="Number of categories")
points(1:m,pr,pch=19)

Probability distribution of a GeCUB model

Description

Compute the probability distribution of a GeCUB model, that is a CUB model with shelter effect with covariates specified for all component.

Usage

probgecub(ordinal,Y,W,X,bet,gama,omega,shelter)

Arguments

Value

A vector of the same length as ordinal, whose i-th component is the probability of the i-th observation according to a GeCUB model with the corresponding values of the covariates for all the components and coefficients specified in bet, gama, omega.

References

Iannario M. and Piccolo D. (2016b). A generalized framework for modelling ordinal data. Statistical Methods and Applications, 25, 163–189.

Probability distribution of an IHG model

Description

Compute the probability distribution of an IHG model (Inverse Hypergeometric) without covariates.

Usage

probihg(m,theta)

Arguments

Value

The vector of the probability distribution of an IHG model.

References

D'Elia A. (2003). Modelling ranks using the inverse hypergeometric distribution, Statistical Modelling: an International Journal, 3, 65–78

Examples

m<-10
theta<-0.30
pr<-probihg(m,theta)
plot(1:m,pr,type="h",xlab="Ordinal categories")
points(1:m,pr,pch=19)

Probability distribution of an IHG model with covariates

Description

Given a vector of n ratings over m categories, it returns a vector of length n whose i-th element is the probability of observing the i-th rating for the corresponding IHG model with parameter \theta_i, obtained via logistic link with covariates and coefficients.

Usage

probihgcovn(m,ordinal,U,nu)

Arguments

Details

The matrix U is expanded with a vector with entries equal to 1 in the first column to include an intercept term in the model.

See Also

probihg

Examples

n<-100
m<-7
theta<-0.30
ordinal<-simihg(n,m,theta)
U<-sample(c(0,1),n,replace=TRUE)
nu<-c(0.12,-0.5)
pr<-probihgcovn(m,ordinal,U,nu)

Relational goods and Leisure time dataset

Description

Dataset consists of the results of a survey aimed at measuring the evaluation of people living in the metropolitan area of Naples, Italy, with respect to of relational goods and leisure time collected in December 2014. Every participant was asked to assess on a 10 point ordinal scale his/her personal score for several relational goods (for instance, time dedicated to friends and family) and to leisure time. In addition, the survey asked respondents to self-evaluate their level of happiness by marking a sign along a horizontal line of 110 millimeters according to their feeling, with the left-most extremity standing for "extremely unhappy", and the right-most extremity corresponding to the status "extremely happy".

Usage

data(relgoods)

Format

The description of subjects' covariates is the following:

ID

An identification number

Gender

A factor with levels: 0 = man, 1 = woman

BirthMonth

A variable indicating the month of birth of the respondent

BirthYear

A variable indicating the year of birth of the respondent

Family

A factor variable indicating the number of members of the family

Year.12

A factor with levels: 1 = if there is any child aged less than 12 in the family, 0 = otherwise

EducationDegree

A factor with levels: 1 = compulsory school, 2 = high school diploma, 3 = Graduated-Bachelor degree, 4 = Graduated-Master degree, 5 = Post graduated

MaritalStatus

A factor with levels: 1 = Unmarried, 2 = Married/Cohabitee, 3 = Separated/Divorced, 4 = Widower

Residence

A factor with levels: 1 = City of Naples, 2 = District of Naples, 3 = Others Campania, 4 = Others Italia, 5 = Foreign countries

Glasses

A factor with levels: 1 = wearing glasses or contact lenses, 0 = otherwise

RightHand

A factor with levels: 1 = right-handed, 0 = left-handed

Smoking

A factor with levels: 1 = smoker, 0 = not smoker

WalkAlone

A factor with levels: 1 = usually walking alone, 0 = usually walking in company

job

A factor with levels: 1 = Not working, 2 = Retired, 3 = occasionally, 4 = fixed-term job, 5 = permanent job

PlaySport

A factor with levels: 1 = Not playing any sport, 2 = Yes, individual sport, 3 = Yes, team sport

Pets

A factor with levels: 1 = owning a pet, 0 = not owning any pet

1. Respondents were asked to evaluate the following items on a 10 point Likert scale, ranging from 1 = "never, at all" to 10 = "always, a lot":

   WalkOut

   How often the respondent goes out for a walk

   Parents

   How often respondent talks at least to one of his/her parents

   MeetRelatives

   How often respondent meets his/her relatives

   Association

   Frequency of involvement in volunteering or different kinds of associations/parties, etc

   RelFriends

   Quality of respondent's relationships with friends

   RelNeighbours

   Quality of the relationships with neighbors

   NeedHelp

   Easiness in asking help whenever in need

   Environment

   Level of comfort with the surrounding environment

   Safety

   Level of safety in the streets

   EndofMonth

   Family making ends meet

   MeetFriend

   Number of times the respondent met his/her friends during the month preceding the interview

   Physician

   Importance of the kindness/simpathy in the selection of respondent's physician

   Happiness

   Each respondent was asked to mark a sign on a 110mm horizontal line according to his/her feeling of happiness (left endpoint corresponding to completely unhappy, right-most endpoint corresponding to extremely happy

2. The same respondents were asked to score the activities for leisure time listed below, according to their involvement/degree of amusement, on a 10 point Likert scale ranging from 1 = "At all, nothing, never" to 10 = "Totally, extremely important, always":

   Videogames

   Reading

   Cinema

   Drawing

   Shopping

   Writing

   Bicycle

   Tv

   StayWFriend

   Spending time with friends

   Groups

   Taking part to associations, meetings, etc.

   Walking

   HandWork

   Hobby, gardening, sewing, etc.

   Internet

   Sport

   SocialNetwork

   Gym

   Quiz

   Crosswords, sudoku, etc.

   MusicInstr

   Playing a musical instrument

   GoAroundCar

   Hanging out by car

   Dog

   Walking out the dog

   GoOutEat

   Go to restaurants/pubs

Details

Period of data collection: December 2014
Mode of collection: questionnaire
Number of observations: 2459
Number of subjects' covariates: 16
Number of analyzed items: 34
Warning: with a limited number of missing values

Simulation routine for CUB models

Description

Generate n pseudo-random observations following the given CUB distribution.

Usage

simcub(n,m,pai,csi)

Arguments

See Also

probcub00

Examples

n<-300
m<-9
pai<-0.4
csi<-0.7
simulation<-simcub(n,m,pai,csi)
plot(table(simulation),xlab="Ordinal categories",ylab="Frequencies")

Simulation routine for CUBE models

Description

Generate n pseudo-random observations following the given CUBE distribution.

Usage

simcube(n,m,pai,csi,phi)

Arguments

See Also

probcube

Examples

n<-300
m<-9
pai<-0.7
csi<-0.4
phi<-0.1
simulation<-simcube(n,m,pai,csi,phi)
plot(table(simulation),xlab="Ordinal categories",ylab="Frequencies")

Simulation routine for CUB models with shelter effect

Description

Generate n pseudo-random observations following the given CUB distribution with shelter effect.

Usage

simcubshe(n,m,pai,csi,delta,shelter)

Arguments

See Also

probcubshe1, probcubshe2, probcubshe3

Examples

n<-300
m<-9
pai<-0.7
csi<-0.3
delta<-0.2
shelter<-3
simulation<-simcubshe(n,m,pai,csi,delta,shelter)
plot(table(simulation),xlab="Ordinal categories",ylab="Frequencies")

Simulation routine for CUSH models

Description

Generate n pseudo-random observations following the distribution of a CUSH model without covariates.

Usage

simcush(n,m,delta,shelter)

Arguments

See Also

probcush

Examples

n<-200
m<-7
delta<-0.3
shelter<-3
simulation<-simcush(n,m,delta,shelter)
plot(table(simulation),xlab="Ordinal categories",ylab="Frequencies")

Simulation routine for IHG models

Description

Generate n pseudo-random observations following the given IHG distribution.

Usage

simihg(n,m,theta)

Arguments

See Also

probihg

Examples

n<-300
m<-9
theta<-0.4
simulation<-simihg(n,m,theta)
plot(table(simulation),xlab="Number of categories",ylab="Frequencies")

S3 method: summary for class "GEM"

Description

S3 method summary for objects of class GEM.

Usage

## S3 method for class 'GEM'
summary(object, correlation = FALSE, ...)

Arguments

Value

Extended summary results of the fitting procedure, including parameter estimates, their standard errors and Wald statistics, maximized log-likelihood compared with that of the saturated model and of a Uniform sample. AIC, BIC and ICOMP indeces are also displayed for model selection. Execution time and number of exectued iterations for the fitting procedure are aslo returned.

Examples

model<-GEM(Formula(MeetRelatives~0|0|0),family="cube",data=relgoods) 
summary(model,correlation=TRUE,digits=4)

Evaluation of the Orientation Services 2002

Description

A sample survey on students evaluation of the Orientation services was conducted across the 13 Faculties of University of Naples Federico II in five waves: participants were asked to express their ratings on a 7 point scale (1 = "very unsatisfied", 7 = "extremely satisfied"). Here dataset collected during 2002 is loaded.

Usage

data(univer)

Format

The description of subjects' covariates is:

Faculty

A factor variable, with levels ranging from 1 to 13 indicating the coding for the different university faculties

Freqserv

A factor with levels: 0 = for not regular users, 1 = for regular users

Age

Variable indicating the age of the respondent in years

Gender

A factor with levels: 0 = man, 1 = woman

Diploma

A factor with levels: 1 = classic studies, 2 = scientific studies, 3 = linguistic, 4 = Professional, 5 = Technical/Accountancy, 6 = others

Residence

A factor with levels: 1 = city NA, 2 = district NA, 3 = others

ChangeFa

A factor with levels: 1 = changed faculty, 2 = not changed faculty

Analyzed ordinal variables (Likert ordinal scale): 1 = "extremely unsatisfied", 2 = "very unsatisfied", 3 = "unsatisfied", 4 = "indifferent", 5 = "satisfied", 6 = "very satisfied", 7 = "extremely satisfied"

Informat

Level of satisfaction about the collected information

Willingn

Level of satisfaction about the willingness of the staff

Officeho

Judgment about the Office hours

Competen

Judgement about the competence of the staff

Global

Global satisfaction

Details

Period of data collection: 2002
Mode of collection: questionnaire
Number of observations: 2179
Number of subjects' covariates: 7
Number of analyzed items: 5

Variance-covariance matrix of a CUB model without covariates

Description

Compute the variance-covariance matrix of parameter estimates of a CUB model without covariates.

Usage

varcovcub00(m, ordinal, pai, csi)

Arguments

Details

The function checks if the variance-covariance matrix is positive-definite: if not, it returns a warning message and produces a matrix with NA entries.

References

Piccolo D. (2006), Observed Information Matrix for MUB Models. Quaderni di Statistica, 8, 33–78,

See Also

probcub00

Examples

data(univer)
m<-7
ordinal<-univer[,12]
pai<-0.87
csi<-0.17
varmat<-varcovcub00(m, ordinal, pai, csi)

Variance-covariance matrix of CUB models with covariates for the feeling component

Description

Compute the variance-covariance matrix of parameter estimates of a CUB model with covariates for the feeling component.

Usage

varcovcub0q(m, ordinal, W, pai, gama)

Arguments

Details

The function checks if the variance-covariance matrix is positive-definite: if not, it returns a warning message and produces a matrix with NA entries.

References

Piccolo D.(2006), Observed Information Matrix for MUB Models. Quaderni di Statistica, 8, 33–78,

Examples

data(univer)
m<-7
ordinal<-univer[,9]
pai<-0.86
gama<-c(-1.94, -0.17)
W<-univer[,4]           
varmat<-varcovcub0q(m, ordinal, W, pai, gama)

Variance-covariance matrix of a CUBE model with covariates

Description

Compute the variance-covariance matrix of parameter estimates of a CUBE model with covariates for all the three parameters.

Usage

varcovcubecov(m, ordinal, Y, W, Z, estbet, estgama, estalpha)

Arguments

Details

The function checks if the variance-covariance matrix is positive-definite: if not, it returns a warning message and produces a matrix with NA entries.

References

Piccolo, D. (2014), Inferential issues on CUBE models with covariates, Communications in Statistics - Theory and Methods, 44, DOI: 10.1080/03610926.2013.821487

Variance-covariance matrix for CUBE models based on the expected information matrix

Description

Compute the variance-covariance matrix of parameter estimates as the inverse of the expected information matrix for a CUBE model without covariates.

Usage

varcovcubeexp(m, pai, csi, phi, n)

Arguments

Details

The function checks if the variance-covariance matrix is positive-definite: if not, it returns a warning message and produces a matrix with NA entries.

References

Iannario, M. (2014). Modelling Uncertainty and Overdispersion in Ordinal Data, Communications in Statistics - Theory and Methods, 43, 771–786

See Also

varcovcubeobs

Variance-covariance matrix for CUBE models based on the observed information matrix

Description

Compute the variance-covariance matrix of parameter estimates for a CUBE model without covariates as the inverse of the observed information matrix.

Usage

varcovcubeobs(m, pai, csi, phi, freq)

Arguments

Details

The function checks if the variance-covariance matrix is positive-definite: if not, it returns a warning message and produces a matrix with NA entries.

References

Iannario, M. (2014). Modelling Uncertainty and Overdispersion in Ordinal Data, Communications in Statistics - Theory and Methods, 43, 771–786

See Also

varcovcubeexp

Variance-covariance matrix of CUB model with covariates for the uncertainty parameter

Description

Compute the variance-covariance matrix of parameter estimates of a CUB model with covariates for the uncertainty component.

Usage

varcovcubp0(m, ordinal, Y, bet, csi)

Arguments

Details

The function checks if the variance-covariance matrix is positive-definite: if not, it returns a warning message and produces a matrix with NA entries.

References

Piccolo D. (2006), Observed Information Matrix for CUB Models, Quaderni di Statistica, 8, 33–78

See Also

probcubpq

Variance-covariance matrix of a CUB model with covariates for both uncertainty and feeling

Description

Compute the variance-covariance matrix of parameter estimates of a CUB model with covariates for both the uncertainty and the feeling components.

Usage

varcovcubpq(m, ordinal, Y, W, bet, gama)

Arguments

Details

The function checks if the variance-covariance matrix is positive-definite: if not, it returns a warning message and produces a matrix with NA entries.

References

Piccolo D. (2006), Observed Information Matrix for CUB Models, Quaderni di Statistica, 8, 33–78

See Also

probcubpq

Variance-covariance matrix for CUB models with shelter effect

Description

Compute the variance-covariance matrix of parameter estimates of a CUB model with shelter effect.

Usage

varcovcubshe(m, pai1, pai2, csi, shelter, n)

Arguments

Details

The function checks if the variance-covariance matrix is positive-definite: if not, it returns a warning message and produces a matrix with NA entries.

References

Iannario, M. (2012), Modelling shelter choices in ordinal data surveys. Statistical Modelling and Applications, 21, 1–22

See Also

probcubshe1

Variance-covariance matrix of a CUB model without covariates

Description

Compute the variance-covariance matrix of parameter estimates of a CUB model without covariates.

Usage

varcovgecub(ordinal,Y,W,X,bet,gama,omega,shelter)

Arguments

Details

The function checks if the variance-covariance matrix is positive-definite: if not, it returns a warning message and produces a matrix with NA entries.

See Also

probgecub

Variance of CUB models without covariates

Description

Compute the variance of a CUB model without covariates.

Usage

varcub00(m,pai,csi)

Arguments

References

Piccolo D. (2003). On the moments of a mixture of uniform and shifted binomial random variables. Quaderni di Statistica, 5, 85–104

See Also

expcub00, probcub00

Examples

m<-9
pai<-0.6
csi<-0.5
varcub<-varcub00(m,pai,csi)

Variance of CUBE models without covariates

Description

Compute the variance of a CUBE model without covariates.

Usage

varcube(m,pai,csi,phi)

Arguments

References

Iannario, M. (2014). Modelling Uncertainty and Overdispersion in Ordinal Data, Communications in Statistics - Theory and Methods, 43, 771–786

See Also

probcube, expcube

Examples

m<-7
pai<-0.8
csi<-0.2
phi<-0.05
varianceCUBE<-varcube(m,pai,csi,phi)

Variance-covariance matrix for CUB models

Description

Compute the variance-covariance matrix of parameter estimates for CUB models with or without covariates for the feeling and the uncertainty parameter, and for extended CUB models with shelter effect.

Usage

varmatCUB(ordinal,m,param,Y=0,W=0,X=0,shelter=0)

Arguments

Details

The function checks if the variance-covariance matrix is positive-definite: if not, it returns a warning message and produces a matrix with NA entries. No missing value should be present neither for ordinal nor for covariate matrices: thus, deletion or imputation procedures should be preliminarily run.

References

Piccolo D. (2006). Observed Information Matrix for MUB Models, Quaderni di Statistica, 8, 33–78
Iannario, M. (2012). Modelling shelter choices in ordinal data surveys. Statistical Modelling and Applications, 21, 1–22
Iannario M. and Piccolo D. (2016b). A generalized framework for modelling ordinal data. Statistical Methods and Applications, 25, 163–189.

See Also

vcov, cormat

Examples

data(univer)
m<-7
### CUB model with no covariate
pai<-0.87; csi<-0.17 
param<-c(pai,csi)
varmat<-varmatCUB(univer$global,m,param)
#######################
### and with covariates for feeling
data(univer)
m<-7
pai<-0.86; gama<-c(-1.94,-0.17)
param<-c(pai,gama)
ordinal<-univer$willingn; W<-univer$gender      
varmat<-varmatCUB(ordinal,m,param,W)
#######################
### CUB model with uncertainty covariates
data(relgoods)
m<-10
naord<-which(is.na(relgoods$Physician))
nacov<-which(is.na(relgoods$Gender))
na<-union(naord,nacov)
ordinal<-relgoods$Physician[-na]
Y<-relgoods$Gender[-na]
bet<-c(-0.81,0.93); csi<-0.20
varmat<-varmatCUB(ordinal,m,param=c(bet,csi),Y=Y)
#######################
### and with covariates for both parameters
data(relgoods)
m<-10
naord<-which(is.na(relgoods$Physician))
nacov<-which(is.na(relgoods$Gender))
na<-union(naord,nacov)
ordinal<-relgoods$Physician[-na]
W<-Y<-relgoods$Gender[-na]
gama<-c(-0.91,-0.7); bet<-c(-0.81,0.93)
varmat<-varmatCUB(ordinal,m,param=c(bet,gama),Y=Y,W=W)
#######################
### Variance-covariance for a CUB model with shelter
m<-8; n<-300
pai1<-0.5; pai2<-0.3; csi<-0.4
shelter<-6
pr<-probcubshe1(m,pai1,pai2,csi,shelter)
ordinal<-sample(1:m,n,prob=pr,replace=TRUE)
param<-c(pai1,pai2,csi)
varmat<-varmatCUB(ordinal,m,param,shelter=shelter)

Variance-covariance matrix for CUBE models

Description

Compute the variance-covariance matrix of parameter estimates for CUBE models when no covariate is specified, or when covariates are included for all the three parameters.

Usage

varmatCUBE(ordinal,m,param,Y=0,W=0,Z=0,expinform=FALSE)

Arguments

Details

The function checks if the variance-covariance matrix is positive-definite: if not, it returns a warning message and produces a matrix with NA entries. No missing value should be present neither for ordinal nor for covariate matrices: thus, deletion or imputation procedures should be preliminarily run.

References

Iannario, M. (2014). Modelling Uncertainty and Overdispersion in Ordinal Data, Communications in Statistics - Theory and Methods, 43, 771–786
Piccolo D. (2015). Inferential issues for CUBE models with covariates, Communications in Statistics. Theory and Methods, 44(23), 771–786.

See Also

vcov, cormat

Examples

m<-7; n<-500
pai<-0.83; csi<-0.19; phi<-0.045
ordinal<-simcube(n,m,pai,csi,phi)
param<-c(pai,csi,phi)
varmat<-varmatCUBE(ordinal,m,param)
##########################
### Including covariates
data(relgoods)
m<-10
naord<-which(is.na(relgoods$Tv))
nacov<-which(is.na(relgoods$BirthYear))
na<-union(naord,nacov)
age<-2014-relgoods$BirthYear[-na]
lage<-log(age)-mean(log(age))
Y<-W<-Z<-lage
ordinal<-relgoods$Tv[-na]
estbet<-c(0.18,1.03); estgama<-c(-0.6,-0.3); estalpha<-c(-2.3,0.92)
param<-c(estbet,estgama,estalpha)
varmat<-varmatCUBE(ordinal,m,param,Y=Y,W=W,Z=Z,expinform=TRUE)

S3 method vcov() for class "GEM"

Description

S3 method: vcov for objects of class GEM.

Usage

## S3 method for class 'GEM'
vcov(object, ...)

Arguments

Value

Variance-covariance matrix of the final ML estimates for parameters of the fitted GEM model. It returns the square of the estimated standard error for CUSH and IHG models with no covariates.

See Also

varmatCUB, varmatCUBE, GEM